Activity pubs.acs.org/jchemeduc
A Tasty Approach to Statistical Experimental Design in High School Chemistry: The Best Lemon Cake Lucia Liguori* Nordahl Grieg High School, Rådal, Bergen NO-5239, Norway S Supporting Information *
ABSTRACT: High school students in a chemistry course were challenged to prepare an improved lemon cake by means of statistical experimental design. They were introduced to basic statistical experimental design theory, calculation of effects, empirical mathematical models, response surfaces and their isocontour projections, as well as descriptive statistics. Geogebra, a digital program for teaching and learning mathematics, was used to produce the mathematical models and response surface plots. Assessment of the lemon cakes with regard to fluffiness (texture), lemon taste, and an overall rating required efficient planning of sensory panels. A slight extrapolation, that is, selection of the experimental setting outside the investigated domain, and assessment of the predicted procedure to check the reproducibility of the lemon cake recipe were also included in this project, which complements the Next Generation Science Standard. KEYWORDS: High School/Introductory Chemistry, First-Year Undergraduate/General, Chemoinformatics, Interdisciplinary/Multidisciplinary, Hands-On Learning/Manipulatives, Problem Solving/Decision Making, Acids/Bases, Chemometrics
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INTRODUCTION
EXPERIMENTAL DESIGN IN PRACTICE The high school students were given a theoretical introduction to the methodology of statistical experimental design. They learned about setting up a 2n factorial experimental design, calculating mathematical models from responses, as well as drawing and interpreting response surfaces in order to optimize a procedure. This paper will focus on implementation of experimental design in high school. Readers interested in the theoretical aspects of experimental design, multivariate analysis, and response surface methodology should be referred elsewhere.19−22 Food and especially sweets always piques students’ interest, so in this case the practical problem was to prepare the best lemon cake. The adjective best in this context signifies a soft and fluffy cake with a distinct taste of lemon. These qualities are unfortunately not easily reproducible. Thus, the students should evaluate and select the parameters that might affect the product quality. The basic recipe that the students used included in total eight various parameters, namely, flour, sugar, eggs, butter, baking powder, lemon juice, the cooking temperature for the cake, and baking time. With a classical approach based on varying one variable at a time, a huge number of experiments should be performed, and no variable interferences can be detected. Through the experimental design approach students identified the most probable variables
Statistical experimental design is fundamental in any optimization process. It helps identify the most influential variables and create an empirical mathematical model suitable to process optimization. Statistical experimental design finds application in research and development, being a commonly used technique in several fields. In analytical biology1 it has been used for DNA purification processes and analysis of peptides. In environmental chemistry,1 it is relevant for determination of contaminant, and in pharmaceutics1 to predict human drug absorption. In food and industrial processes1 the method helps to increase antioxidant extraction from nuts2,3 and seeds,4 and to monitor acrylamide levels in fried potato crisps.5 Moreover, statistical experimental design is important in wine production6,7 and investigation of consumer perceptions of organic wine.8 In chemistry education, statistical experimental design has been used in graduate and upper-division undergraduate groups in several disciplines such as analytical quantitative chemistry,9−13 organic14 and organometallic chemistry,15 and biology.16,17 To the best of my knowledge, there is no literature on the application of statistical experimental design in the high school chemistry laboratory. The herein reported lemon cake project is in line with the Next Generation Science Standards (NGSS)18 learning requirements. In fact, the project is a multidisciplinary experiment example involving chemistry, multivariate mathematics, experimental design, and dynamic mathematics software Geogebra. © XXXX American Chemical Society and Division of Chemical Education, Inc.
Received: June 21, 2016 Revised: January 10, 2017
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DOI: 10.1021/acs.jchemed.6b00369 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Activity
influencing the process yield, namely, the desired cake quality. Two variables were then selected: the quantity of baking powder (x1) determining fluffiness and smoothness, and the quantity of lemon juice (x2) determining the flavor. Students recognized that these two variables, x1 and x2, are strongly correlated due to the chemical acid−base reaction between citric acid C6H8O7 in the lemon juice and NaHCO3 in the baking powder23 (eq 1).
Table 2. Sensory Panel Organization sensory board
3NaHCO3 + C6H8O7 → C6H5O7 Na3 + 3CO2 + 3H 2O (1)
In order to find a good lemon cake recipe, two preliminary experiments were conducted by using two recipes differing in both procedure and quantity of baking powder and lemon juice. In the first recipe, baking powder was added directly to the lemon juice. In the second recipe, a lesser amount of baking powder was added to the flour, while a bigger quantity of lemon juice was carefully added to the egg whites during the mounting process (details on the preliminary recipes are in the Supporting Information). The resulting cakes were reasonably different in lemon taste and leavening degree. Although the first cake had a higher quantity of baking powder, it was flat and thick. Moreover, the lemon taste was barely noticeable. The second one was spongy, but disliked because of the strong lemon flavor. The class decided to use a recipe that was a middle ground between the first and the second recipe. For the procedure, the class selected the second recipe, avoiding a direct reaction between baking powder and lemon juice. In the experimental design, the baking powder varied between 16 and 24 g, coded levels −1 and +1. The quantity of the lemon juice varied between 84 and 150 g, coded levels −1 and +1. Coded level (0, 0) that corresponded to 20 g baking powder and 117 g lemon juice (Table 1) represents the center point experiment in the
a
−1
0
+1
qty baking powder [g] qty lemon juice [g]
x1 x2
16 84
20 117
24 150
A B (Ar) C D (Cr) E F (Er) G H (Gr) J I (Jr)
X
2
3
4
5
X
6
7
X
X
X X
X
X
X X
X X
10 X X
X X
X X X
9
X X X
X
X X
8 X
X X
X
X
X
X
X X
X
X
X X
X
X
Cake Z (Yr) means Z is replicate of cake Y.
with j = 1, 2, 3 and x = x1 , x 2 , x12 E1(x1) =
(2)
1 1 (3.06 + 2.79) − (3.61 + 3.08) = −0.42 2 2 (2a)
Similar calculations are reported in the Supporting Information and can be run for variable x2 and interaction x1x2. Table 3 shows the values of all of the effects. Table 3. Factorial Effects in the Lemon Cake 22 Factorial Design
experimental levels xi
1
According to Yates25 definition, the difference in the level of response (y,̅ mean value) moving from the low (−1) to the high level (+1) represents the main effect of that variable (eq 2). For example, the main effect E1(x1) of variable x1 is shown by eq 2a. 1 Ej(x) = (yj(̅ +1) − yj(̅ −1) ) 2
Table 1. Experimental Variables and Levels exptl variable
cakea
a
selected experimental domain defined by the lower (−1, −1) and upper (+1, +1) levels. The inclusion of the center point in our experimental design was important to check if the selection of the experimental domain levels (−1, −1) and (+1, +1) gave enough variation in the responses. Moreover, the center point experiment is always recommended22 in order to assess the response surface (presence of quadratic effects) and to perform any statistical analysis, especially for more complicated mathematical models. The organization of the taste panels was not a simple task,24 but students were challenged to make a plan. After discussion, it was decided to have 10 tasting stations, each visited by 10 students from other classes. At each station, students tasted four pieces of cake, of which three were from different cakes and one was a replicate. In this way, 100 students attended the taste panels and each cake was evaluated 40 times (Table 2). Students judged lemon taste (y1), fluffiness (y2), and overall rating (y3) of the cake using a rating scale from 1, very bad, to 6, excellent. The scale choice 1−6 is the same as the school grading scale to which students are accustomed.
responsea
Ej(x1)
Ej(x2)
Ej(x1x2)
mean
y1 y2 y3
−0.420 −0.520 −0.234
−0.400 −0.180 −0.505
0.130 −0.020 0.156
3.154 3.420 3.346
y1, lemon taste; y2, fluffiness; y3, overall rating.
For each of the three responses (lemon taste y1, fluffiness y2, and overall rating y3), a general equation for the multivariate mathematical model is eq 3 2
yj = βj + 0
∑ βj xi + βj i=1
i
i
